TABLE OF CONTENTS As in the previous chapter, we conclude this chapter with numerical examples that illustrate the behavior of some of the iterative regularization methods, with special focus on regularizing CG iterations. The test problem used through out the examples is again shaw from 1.4.3, discretized with m = n = 64.
The error history for an iterative method with iterates x ( k) is a plot of the relative error norm ? x exact ? x ( k) ? 2/ ? x exact ? 2 versus the iteration number k. Figure 6.7 shows the error histories for classical Landweber iteration (6.2) with ? = 0.15, the v-method (6.7), Kaczmarz's method (6.10), and CGLS (6.14) applied to the shaw test problem with exact right-hand side. Notice the nonmonotonic behavior of the error for the v-method.
All four methods have an initial stage in which the convergence is faster than in the remaining iterations. In this initial stage, the solution's SVD components corresponding to the largest singular values are picked up. Overall, the first three methods converge much more slowly than CGLS, and the example clearly illustrates the potential power of regularizing...
